[[1]}]
The map \(\psi Z(m,n) \rightarrow \mathbf {k}[\mathfrak {h}]\)  is a monomorphism and the image consists of \((S_m \times S_n)\) -invariant functions \(f\)  such that
\(f(\lambda ) = f(\lambda + t \alpha )\) 
for any \(t \in \mathbf {k}\) , whenever \(\lambda \in H_\alpha \)  for an isotropic root \(\alpha \) .
